Abstract: There has been extensive interest in recent years in the application of high-throughput microfluidic systems to crystallize a variety of organic compounds including amino acids, proteins, and active pharmaceutical ingredients (APIs) (Hansen et al, 2006; Sanjoh & Tsukihara, 1999; Talreja et al, 2005; Zheng et al, 2003). The goals of such studies include the identification of crystallization conditions to produce a single protein crystal for structure determination, the high-throughput search for polymorphs for pharmaceutical compounds, and the exploration of the behavior of crystallization kinetics at high supersaturation. This presentation reports recent progress in formulating a suitable model for estimating the nucleation rate expression at high supersaturations using experimental data obtained from evaporation-based microfluidic crystallization (Talreja et al, 2005).
Due to the small volumes, the measured induction times for crystallization within microfluidic systems are stochastic, and deterministic models as used by some researchers (e.g., Talreja et al, 2007) to estimate nucleation kinetics for these systems fail to capture this randomness. Stochastic models for nucleation at small length scales that have been developed for the case of constant supersaturation created by varying temperature (Galkin & Vekilov, 1999) do not apply for evaporation-based microfluidic crystallization in which the supersaturation varies during the experiment. An advantage of using evaporation-based microfluidic crystallization is that for pharmaceutical systems each experiment nearly always produces a crystal and hence a data point (the induction time), without any prior knowledge of the nucleation kinetics.
More generally, stochastic models have been developed to analyze the behavior of particles specified in terms of probability functions. These include solving for fluctuations about the expected value using higher order product density equations by Ramkrishna (1973; 1974) and using the interval of quiescence---a random variable used to represent the length of time that elapses after a particulate phenomenon takes place and before the next one---to model the probability distribution for different types of phenomena by Shah et al (1977). This presentation reports the Master equation for evaporation-based microfluidic crystallization and its direct analytical solution which is systematically derived from a variant of biological population growth processes (Kendall, 1949). The Master equation is a set of first-order differential equations describing the time-evolution of the probability of the system, which is solved by comparing coefficients with a probability generating function. The solution to these equations generate the probability that the number of crystals in the system, N(t), has the value n at any time t. An expression is derived for the mean induction time and the most likely induction time, which is shown to be equal.
Using this model and experimental data from He et al (2006), a detailed analysis is carried to determine the shape of the nucleation expression rate for paracetamol using various numerical methods. The experimental data are induction times obtained over a wide range of initial concentrations and evaporation rates. In this presentation, it is shown that the measured induction times for the full range of initial concentrations and evaporation rates are not well fit by the standard power-law nucleation rate expression or the classical nucleation model (e.g., Oxtoby, 1992). This motivates the use of a basis function expansion of the nucleation rate expression to obtain an understanding of its dependency on the supersaturation, and an identifiability analysis to assess how much information can be obtained on the form of the nucleation rate expression from the available experimental data. While the nucleation rate expression is not well described by low-order polynomial basis functions, a nucleation rate expression based on piecewise polynomial basis functions fit the data using a least-squares algorithm with penalty. Uncertainties in the nucleation kinetic expressions are carefully quantified throughout the analyses.
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